Spinor analysis on C² and on conformally flat 4-manifolds

Abstract

Abstract.The present paper is concerned with extensions of complex analysis on the complex plane C to conformally flat 4-manifolds. We shall give in an explicit form a fundamental system of spinors that will serve as the basis vectors for the Laurent expansion. Restricted to a sphere around the center of the expansion these spinors form a complete orthonormal system of eigenspinors of the tangential Dirac operator on the sphere, and give a basis of the representations of Spin(4). We shall also give the definition of meromorphic spinors and residues, and prove under some hypothesis that, on a compact conformally flat 4-manifold, the sum of the residues of a meromorphic spinor is zero.

Quaternifications and Extensions of Current Algebras on S3

“…In this section we shall review the analysis of the Dirac operator on H ≃ C 2 . The general references are [B-D-S] and [G-M], and we follow the calculations developed in [Ko1], [Ko2] and [Ko3].…”

“…In Section 2 we shall review the theory of spinor analysis after [8,9]. Let D : ; m = 0, 1, · · · }, with multiplicity (m + 1)(m + 2).…”

Current algebras on S^3 of complex Lie algebras

“…Then we shall introduce a triple of 2-cocycles on L , and extend them to 2-cocycles on the current algebra Lg. For this purpose we prepare in section 2 a rather long introduction to our previous results on analysis of harmonic spinors on C 2 , [F,Ko1,Ko2,Ko3] and [K-I], that is, we develop some parallel results as in classical analysis; the separation of variable method for Dirichlet problem, Fourier expansion by the eigenfuctions of Laplacian, Cauchy integral formula for holomorphic functions and Laurent expansion of meromorphic functions etc.. For example, the Dirac operator on spinors corresponds to the Cauchy-Riemann operator on complex functions.…”

“…2 , − m+3 2 ; m = 0, 1, • • • } with multiplicity (m + 1)(m + 2). We have an explicitly written polynomial formula of eigenspinors φ +(m,l,k) , φ −(m,l,k) 0≤l≤m, 0≤k≤m+1 corresponding to the eigenvalues m 2 and − m+3 2 respectively that give rise to a complete orthonormal system in L 2 (S 3 , S + ), [Ko1,Ko2]. A spinor φ on a domain G ⊂ C 2 is called a harmonic spinor on G if Dφ = 0.…”

“…Here we prepare a fairly long preliminary of spinor analysis on R 4 because I think various subjects belonging to quaternion analysis or detailed properties of harmonic spinors of the Dirac operator on R 4 are not so familiar to the readers. We refer to [F,Ko1] for the exposition on Dirac operators on R 4 and to Ko2] for the function theory of harmonic spinors. Subsections 2.1, 2.2, 2.3 are to remember the theory of harmonic spinors.…”